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Theorem ialgrlem1st 11451
Description: Lemma for ialgr0 11453. Expressing algrflemg 6033 in a form suitable for theorems such as seq3-1 10014 or seqf 10015. (Contributed by Jim Kingdon, 22-Jul-2021.)
Hypothesis
Ref Expression
ialgrlem1st.f (𝜑𝐹:𝑆𝑆)
Assertion
Ref Expression
ialgrlem1st ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥(𝐹 ∘ 1st )𝑦) ∈ 𝑆)

Proof of Theorem ialgrlem1st
StepHypRef Expression
1 algrflemg 6033 . . 3 ((𝑥𝑆𝑦𝑆) → (𝑥(𝐹 ∘ 1st )𝑦) = (𝐹𝑥))
21adantl 272 . 2 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥(𝐹 ∘ 1st )𝑦) = (𝐹𝑥))
3 ialgrlem1st.f . . . 4 (𝜑𝐹:𝑆𝑆)
43adantr 271 . . 3 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → 𝐹:𝑆𝑆)
5 simprl 499 . . 3 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → 𝑥𝑆)
64, 5ffvelrnd 5474 . 2 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝐹𝑥) ∈ 𝑆)
72, 6eqeltrd 2171 1 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥(𝐹 ∘ 1st )𝑦) ∈ 𝑆)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1296  wcel 1445  ccom 4471  wf 5045  cfv 5049  (class class class)co 5690  1st c1st 5947
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060  ax-un 4284
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635  df-sbc 2855  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-br 3868  df-opab 3922  df-mpt 3923  df-id 4144  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-fo 5055  df-fv 5057  df-ov 5693  df-1st 5949
This theorem is referenced by:  ialgr0  11453  algrp1  11455
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